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Industrial Engineering

Traditional Inventory Models for Independent Demand. Industrial Engineering. Prof. Prem Vrat Mechanical Engineering Dept. IIT Delhi. What is an Inventory System. Inventory is defined as the stock of any item or resource used in an organization.

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Industrial Engineering

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  1. Traditional Inventory Models for Independent Demand Industrial Engineering Prof. Prem Vrat Mechanical Engineering Dept. IIT Delhi

  2. What is an Inventory System • Inventory is defined as the stock of any item or resource used in an organization. • An Inventory System is made up of a set of policies and controls designed to monitor the levels of inventory and designed to answer the following questions: • What levels should be maintained? • When stock should be replenished? and • How large orders should be? i.e. what is the optimal size of the order?

  3. Cost of holding inventory • Capital cost (Interest) • Storage cost • Obsolescence cost • Consist of about 20% of total cost in the United States

  4. Inventory issues • Demand • Constant vs. variable • deterministic vs. stochastic • Lead time • Review time • Continuous vs. periodic • Excess demand • Backorders, lost sales • Inventory change • Perish, obsolescence Inventory Decisions: When, What, and how many to order

  5. Two basic types of Inventory Systems 1) continuous (fixed-order quantity) • an order is placed for the same constant amount when inventory decreases to a specified level, ie. Re-order point 2) periodic (fixed-time) • an order is placed for a variable amount after a specified period of time • used in smaller retail stores, drugstores, grocery stores and offices

  6. basic inventory elements • Carrying cost, Cc • Include facility operating costs, record keeping, interest, etc. • Ordering cost, Co • Include purchase orders, shipping, handling, inspection, etc. • Shortage (stock out) cost, Cs • Sometimes penalties involved; if customer is internal, work delays could result

  7. Carrying Costs

  8. Inventory - to study methods to deal with “how much stock of items should be kept on hands that would meet customer demand” Objectives are to determine: a) how much to order, and b) when to order

  9. Inventory models Here, we only study the following three different models: 1. Basic model 2. Model with “discount rate” 3. Model with “re-order points”

  10. 1. Basic model The basic model is known as: “Economic Order Quantity” (EOQ) Models Objective is to determine the optimal order size that will minimize total inventory costs How the objective is being achieved?

  11. Profile of Inventory Level Over Time Q Usage rate Quantity on hand Reorder point Receive order Place order Place order Receive order Receive order Lead time

  12. Profile of … Frequent Orders

  13. Basic EOQ models Three models to be discussed: • Basic EOQ model • EOQ model without instantaneous receipt 3. EOQ model with shortages.

  14. Basic Fixed-Order Quantity Model • This model attempts to estimate the order size (Q) and determine the point (R) at which an order should be placed. • Model assumptions: • Annual demand (D) for the product is known, constant and uniform throughout the period, • Lead time (L) is known and constant, • Product unit price (C) is known and constant, • Per unit holding or carrying cost (Cc) is known and constant, • Ordering or setup cost (Co) is known and constant,

  15. Basic Fixed-Order Quantity Model Model assumptions: 6 No backorders are allowed, 7. There is no interaction with other products, the inventory control system operates independent of its environment.

  16. 1 Economic Order Quantity (EOQ) Model Q Q/2 Order Size R L L Time d Daily Usage Rate

  17. The Basic EOQ Model • The optimal order size, Q, is to minimize the sum of carrying costs and ordering costs. • Assumptions and Restrictions: - Demand is known with certainty and is relatively constant over time. - No shortages are allowed. - Lead time for the receipt of orders is constant. (will consider later) - The order quantity is received all at once and instantaniously. How to determine the optimal value Q*?

  18. Determine of Q We try to • Find the total cost that need to spend for keeping inventory on hands • = total ordering + stock on hands • Determine its optimal solution by finding its first derivative with respect to Q How to get these values? • Find out the total carrying cost • Find out the total ordering cost • Total cost = 1 + 2 • d (Total cost) /d Q = 0, and find Q*

  19. The Basic EOQ Model We assumed that, we will only keep half the inventory over a year then The total carry cost/yr = Cc x (Q/2). Total order cost = Co x (D/Q) Finding optimal Q* Then , Total cost =

  20. The Basic EOQ Model • EOQ occurs where total cost curve is at minimum value and carrying cost equals ordering cost: • Where is Q* located in our model? * Then, (How to obtain this?)

  21. EOQ Derivation Min. The Total Cost Function by finding first derivative and equating it to zero: TC = DCo/Q + QCc/2 dTC Solving for Q: EOQ = We could achieve the same result by equating Holding (Carrying Cost) to Ordering Cost and solve for Q . Reorder point: R = d.L + SS (safety stock) 2 = (- DCo/Q ) + Cc/2 = 0 dQ 2DCo Cc

  22. To order inventory To keep inventory The Basic EOQ Model • Total annual inventory cost is sum of ordering and carrying cost: Figure The EOQ cost model Try to get this value

  23. The Basic EOQ ModelExample Consider the following: * Note: You should pay attention that all measurement units must be the same Consider the same example, with yearly No. of working days/yr

  24. The Basic EOQ ModelEOQ Analysis with monthly time frame (unit be based on yearly) 12 months a year

  25. Robust Model • The EOQ model is robust • It works even if all parameters and assumptions are not met • The total cost curve is relatively flat in the area of the EOQ

  26. 2 Fixed-Order Quantity Model With Usage In this model production and usage of the item being manufactured occur simultaneously. The graph below illustrates the model. TC = DC + DS/Q + Production Rate, p Max. No production usage only Q Build up Usage Rate (p-d) R L Usage Rate, d (p - d).Q.H 2p

  27. The EOQ Model with Noninstantaneous Receipt The order quantity is received gradually over time and inventory is drawn on at the same time it is being replenished. Example: Let p = production, d = demand, Figure The EOQ model with noninstantaneous order receipt Always greater than 0 why?

  28. The EOQ Model with Noninstantaneous ReceiptModel Formulation Assuming placing an order/yr

  29. The EOQ Model with Noninstantaneous ReceiptExample Let,

  30. The EOQ Model with Shortages Here, we allow Q being shortage, so that we could borrow or replenish the stocks later Max level of inventory • In the EOQ model wth shortages, the assumption that shortages cannot exist is relaxed. • Assumed that unmet demand can be backordered with all demand eventually satisfied. Shortage = S/Q Shortage What we needed Total cost is On hand = (Q-S)/Q t1 + t2 = S/D + (Q-S)/D = Q/D

  31. 3 The EOQ Model with Shortages Area = ½ * (S/Q) * S = ½ * S2 /Q ½*base* height = ½ * (Q-S) * (Q-S)/Q = ½ * (Q-S)2 /Q • In the EOQ model wth shortages, the assumption that shortages cannot exist is relaxed. • Assumed that unmet demand can be backordered with all demand eventually satisfied. Shortage = S/Q Shortage What we needed On hand = (Q-S)/Q t1 + t2 = S/D + (Q-S)/D = Q/D

  32. The EOQ Model with Shortages

  33. The EOQ Model with ShortagesExample Let,

  34. The EOQ Model with ShortagesAdditional Parameters in Example = Q/D

  35. 4. Model with “discount rate” Price discounts are often offered if a predetermined number of units is ordered or when ordering materials in high volume. How do we decide if we should order more to take advantage of the discount being offered?

  36. All-Unit Quantity Discounts • Pricing schedule has specified quantity break points q0, q1, …, qr, where q0 = 0 • If an order is placed that is at least as large as qi but smaller than qi+1, then each unit has an average unit cost of Ci • The unit cost generally decreases as the quantity increases, i.e., C0>C1>…>Cr • The objective for the company (a retailer for example) is to decide on a lot size that will minimize the sum of material, order, and holding costs

  37. All-Unit Quantity Discount Procedure Step 1: Calculate the EOQ for the lowest price. If it is feasible (i.e., this order quantity is in the range for that price), then stop. This is the optimal lot size. Calculate TC for this lot size. Step 2: If the EOQ is not feasible, calculate the TC for this price and the smallest quantity for that price. Step 3: Calculate the EOQ for the next lowest price. If it is feasible, stop and calculate the TC for that quantity and price. Step 4: Compare the TC for Steps 2 and 3. Choose the quantity corresponding to the lowest TC. Step 5: If the EOQ in Step 3 is not feasible, repeat Steps 2, 3, and 4 until a feasible EOQ is found.

  38. All-Unit Quantity Discounts: Example Cost/Unit Total Material Cost $3 $2.96 $2.92 10,000 10,000 5,000 5,000 Order Quantity Order Quantity

  39. All-Unit Quantity Discount: Example Order quantity Unit Price 0-5000 $3.00 5001-10000 $2.96 Over 10000 $2.92 q0 = 0, q1 = 5000, q2 = 10000 C0 = $3.00, C1 = $2.96, C2 = $2.92 D = 120000 units/year, Co = $100/lot, Cc = 0.2

  40. All-Unit Quantity Discount: Example Step 1: Calculate Q2* = Sqrt[(2DCo)/CcC2] = Sqrt[(2)(120000)(100)/(0.2)(2.92)] = 6410 Not feasible (6410 < 10001) Calculate TC2 using C2 = $2.92 and q2 = 10001 TC2 = (120000/10001)(100)+(10001/2)(0.2)(2.92)+ (120000)(2.92) = $354,520 Step 2: Calculate Q1* = Sqrt[(2DCo)/CcC1] =Sqrt[(2)(120000)(100)/(0.2)(2.96)] = 6367 Feasible (5000<6367<10000)  Stop TC1 = (120000/6367)(100)+(6367/2)(0.2)(2.96)+ (120000)(2.96) = $358,969 TC2 < TC1 The optimal order quantity Q* is q2 = 10001

  41. All-Unit Quantity Discounts • What is the effect of such a discount schedule? • Retailers are encouraged to increase the size of their orders • Average inventory (cycle inventory) in the supply chain is increased • Average flow time is increased • Is an all-unit quantity discount an advantage in the supply chain?

  42. 5. Model with “re-order points” • The reorder point is the inventory level at which a new order is placed. • Order must be made while there is enough stock in place to cover demand during lead time. • Formulation: R = dL, where d = demand rate per time period, L = lead time • Then R = dL = (10,000/311)(10) = 321.54 Working days/yr

  43. Reorder Point • Inventory level might be depleted at slower or faster rate during lead time. • When demand is uncertain, safety stock is added as a hedge against stockout. Two possible scenarios No Safety stocks! Safety stock! We should then ensure Safety stock is secured!

  44. Determining Safety Stocks Using Service Levels • We apply the Z test to secure its safety level, Safety stock Reorder point Average sample demand How these values are represented in the diagram of normal distribution?

  45. Reorder Point with Variable Demand

  46. Reorder Point with Variable DemandExample Example: determine reorder point and safety stock for service level of 95%.

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